An efficient numerical method for predicting the natural frequencies of cylindrical helical springs

In this study, the stiffness method is employed for the free vibration problem of cylindrical helical springs. The element stiffness matrix for the helical spring with twelve degrees-of-freedom is obtained exactly by the transfer matrix method. The efficacious numerical algorithm is employed for the computation of the element transfer matrix. The concentrated element mass matrix is used. The subspace iteration method is preferred for the solution of the large-scale eigenvalue problem. The axial and shear deformation and the rotary inertia terms are considered in the formulation. The free vibrational parameters are chosen as the number of coils (n = 3-16), the helix pitch angle (a = 5-25 degrees), the shape of cross-section (circular, hollow circle and squared) and as the ratio of the diameters of cylinder to wire (D/d = 4-16) in a wide range. Solving the miscellaneous problems, the non-dimensional charts are obtained for the cylindrical helical springs fixed at both ends. Using these charts the natural frequencies are expressed in analytical form in a very good approximation (with the maximum absolute relative error of 5%) and presented for the designers. (C) 1999 Elsevier Science Ltd. All rights reserved.